Editing Exponential distribution (section)

===Conditional Value at Risk (Expected Shortfall)===
The conditional value at risk (CVaR) also known as the [[expected shortfall]] or superquantile for Exp(''λ'') is derived as follows:<ref name="Norton-2019">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer |doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27 |archive-date=2023-03-31 |archive-url=https://web.archive.org/web/20230331230821/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |url-status=dead }}</ref>

<math display="block">\begin{align}
\bar{q}_\alpha (X) &=  \frac{1}{1-\alpha} \int_{\alpha}^{1} q_p (X) dp \\
&=  \frac{1}{(1-\alpha)} \int_{\alpha}^{1} \frac{-\ln (1 - p )}{\lambda}  dp \\
&=  \frac{-1}{\lambda(1-\alpha)} \int_{1-\alpha}^{0} -\ln (y )  dy \\
&=  \frac{-1}{\lambda(1-\alpha)} \int_{0}^{1 - \alpha} \ln (y )  dy \\
&=   \frac{-1}{\lambda(1-\alpha)}  [  ( 1-\alpha) \ln(1-\alpha) - (1-\alpha) ] \\
&=   \frac{ - \ln(1-\alpha) + 1 } { \lambda} \\
\end{align}
</math>